Multiplier apparatus using function generators



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MULTIPLIER APPARATUS USING FUNCTION GENERATORS Original Filed May 11,1959 10 Sheets-Sheet 8 INVENTOR. HUGO M. MART/NE Z A T TORNEY Oct. 25,1966 H. M. MARTINEZ 3,281,584

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A TTO/PNE V Oct. 25, 1966 10 Sheets-Sheet 1O A 7' TOPNEV United StatesPatent 3,281,584 MULTIPLIER APPARATUS USING FUNCTION GENERATORS Hugo M.Martinez, Chicago, Ill., assignorto Yuba Consolidated Industries, Inc.,San Francisco, Calif., a corporation of Delaware Original applicationMay 11, 1959, Ser. No. 812,566. Divided and this application Dec. 10,1962, Ser. No.

1 Claim. (Cl. 235-194) The invention described herein may bemanufactured and used by or for the Government of the United States ofAmerica for governmental purposes without the payment of any royaltiesthereon or therefor.

This application is a division of my copending application Serial No.812,566, filed May 11, 1959.

This invention relates to methods and apparatus for producing physicalquantities respresentative of mathematical functions, and moreparticularly of the product of two independent variables.

Computing devices and other equipment often require the generation ofphysical quantities such as voltages, currents, displacements, or thelike, representative of various mathematical functions. Arrangements foraccomplishing these purposes are commonly known as function generators.Prior art function generators often have been complicated and difficultto construct. An object of the present invention is to providerelatively simple, reliable, and accurate apparatus and methods forgenerating functions, and more particularly functions representing theproduct of two independent variables.

Other objects and many of the attend-ant advantages of this inventionwill be readily appreciated as the same becomes better understood byreference to the following detailed description when considered inconnection with the accompanying drawings wherein:

FIG. 1 is a graph illustrating a periodic time representation of alinear function having a duty cycle less than 100%;

FIG. 2 is a block diagram showing one apparatus of this invention;

FIG. 3 is a graph illustrating a periodic time representation of alinear function having a 100% duty cycle;

FIG. 4 is a graph illustrating a periodic time representation of themonotonic segment of the sine function having a 100% duty cycle;

FIG. 5 is a block diagram of an apparatus for generating the arc sinefunction;

FIG. 6 is a schematic diagram of an apparatus for generating the sineand cosine functions using the apparatus of FIG. 5 in the feedback of anamplifier;

FIG. 7 is a block diagram of an apparatus for generating the sine andcosine functions using the basic method of the invention;

FIG. 8 is a graph illustrating a periodic time representation of the arcsine function produced from a sine wave;

FIG. 9 is a graph illustrating the static function set-up used to modifya sine wave to produce the graph of FIG. 8;

FIG. 10 is a schematic diagram of an apparatus, using an arc sinegenerator in the feedback of an amplifier, for producing the sinefunction and cosine function for a range of angles extending over 31rradians;

FIG. 11 is a graph illustrating the operation of the means in FIG. 10for extending the usable angular range;

FIG. 12 is a schematic diagram of an apparatus using the basic method ofthe invention for generating the sine and cosine functions with meansextending the angular range over 31r radians;

FIG. 13 is a schematic diagram of an apparatus used ice for extendingwithout limit the angular range of sine and cosine generators;

FIG. 14 is a schematic diagram of an apparatus for accomplishing polarto rectangular transformations using arc sine generators in the feedbackof amplifiers;

FIG. 15 is a schematic diagram of an apparatus for accomplishing polartorectangular transformations by direct application of the basic methodof the invention;

FIG. 16 is a schematic diagram of a four-quadrant multiplier;

FIG. 17 is a schematic diagram of an apparatus for generating a periodictime representation of a positive exponential;

FIG. 18 is a schematic diagram of an apparatus using the apparatus ofFIG. 17 in the basic method of the invention for generating alogarithmic function;

FIG. 19 is a schematic diagram of an apparatus using the apparatus ofFIG. 18 in the feedback of an amplifier for generating an exponentialfunction;

FIG. 20 is a schematic diagram of an apparatus for generatingessentially a periodic time representation of a negative exponential;

FIG. 21 is a schematic diagram of an apparatu using the apparatus ofFIG. 20 and the basic method of the invention to generate the logarithmof reciprocals;

FIG. 22 is a schematic diagram of a circuit using the apparatus of FIG.21 in the feedback of an amplifier for producing negative exponentials;

FIG. 23 is a schematic diagram of a circuit using the apparatus of FIG.18 for producing positive constant powers of a variable;

FIG. 24 is a schematic diagram of a circuit using the apparatus of FIG.18 and of FIG. 21 for generating negative constant powers of a variable;

FIG. 25 is a schematic diagram of a circuit using the apapratus of FIG.18 for generating variable powers of a variable;

FIG. 26 is a schematic diagram of an apparatus for producing periodictime representations of a linear function, a quadratic function, a cubicfunction, etc.;

FIG. 27(a) is a schematic diagram of a circuit using the apparatus ofFIG. 26 for generating square roots;

FIG. 27(b) is a schematic diagram of a circuit using the apparatus ofFIG. 26 for generating cube roots;

FIG. 28(a) is a schematic diagram of a circuit using the apparatus ofFIG. 27(a) to generate squares;

FIG. 28(1)) is a schematic diagram of a circuit using the apparatus ofFIG. 27(b) for generating cubes; and

FIG. 29 is a schematic diagram of an apparatus using trigonometricrelations to generate constant powers without the use of logarithms.

The methods and apparatus of the invention are based on what arebelieved to be certain novel mathematical relations. For an adequateunderstanding of the invention, an exposition of these relations isfirst set forth herewith.

A function is a quantity which takes on a definite value, or values,when special values are assigned to certain quantities, called thearguments or independent v-ariables of the function. Examples offunctions of one variable, x ,are the following: 2x;(1x sin x; e; log x.These are also called functional expressions. One quantity is said to bea function of another if to each value of the second (the independentvariable) there corresponds a value of the first (the dependentvariable). The range of the independent variable is either explicitystated, or understood from the context. The foregoing examples offunctional expressions are specific functions of x. The symbols usedfora .general function of x are f(x), g(x), F(x), (x), etc. Such symbolsare used when making statements that are true for several differentfunctions, in

other words, statements that are not concerned with a specific form offunction. Frequently a single symbol, constituting the indepedentvariable, is used to represent a function and is then defined as equalto the particular, specific functional expression in the dependentvariable or to the general function. Thus, for example, Where the symboly is used to rep-resent a function it may, using the previousexpressions as examples, be defined specifically as y=2x; y=(1x y=sin x;etc, or it may be defined in the case of a general function as y=;f(x):y=g( An inverse function or the inverse of a function is the functionobtained by experessing the indepedent variable explicity in terms ofthe dependent variable and considering the dependent variable as anindependent variable. If y=f(x) result-s in x=g(y), the latter is theinverse of the former (and vice versa). Thus where a function y isdefined as y=2x, the inverse function is x= /2 y. In the case of thegeneral function where y=f(x), the inverse function is written x==f-(y).

It must be remembered that a function is always regarded at beingconfined within limits constituting the range of interest. That is,there are limiting values to the function which depend on eitherexplicity expressed limits of the dependent variable or, impliedly,those limits of the dependent variable for which the function isdefined.

A function generator is an apparatus which, assuming the functionalrelation between two variables, for example, to be expressed by y=f(x),will, when supplied with any particular value of x, say x within thelimits of the function produce the corresponding value of y, say y Thisprocess of producing from a given value of x the corresponding value ofy is called generating a function.

Denoting in general a functional relation between two variables byy=f(x) and the inverse function by x=f (y), the method of the presentinvention achieves the automatic physical realization of the relationy=f(x) by the use of the relation x=7' -(y). This means that given aspecific value of x in some physical form such as a voltage, current, orthe like, then the corresponding value of y will be generated in thesame or analogous physical form, using the relation x=f- (y). It isnoted again that while in the relation y=f(x), x is the independentvariable and y is the dependent variable, the reverse is true in theinverse relation x=f (y). As a specific example, if y=arc sin xcorresponds to y=f(x) wherein f(x)=arc sin x, then x=sin y correspondsto =f (y) and f* (y)= y- Prior art automatic generation of functions bythe use of given inverse functions has been accomplished byautomatically solving the equation x-]- (y)=() using y as the unknown.of the book Electronic Analog Computers by G. A. Korn and T. M. Korn,published by McGraw-Hill Book Company, New York, second edition, 1956.The practical success of such equation solving methods is largelydependent on the ease with which f- (y) can be generated. By generationof f (y) is meant that given a value of y, the corresponding value of f(y) is produced. These methods all give static representation of f- (y),wherein y is time independent.

In contrast to the foregoing automatic equation solving method, themethod of the present invention does not rely on the solving ofequations; and instead of a static representation of f- (y) it uses adynamic representation or time representation of (y) by, in effect,replacing y with real time, in which replacement an interval of timerepresents the range of y. To understand this method, an explanation ofcertain terms is appropriate. A time representation of a function :g(x)defined for spond to the range of x from x to x and generating Thissystem is explained on page 340.

the function =g(x) as a function of time over this interval.Specifically, a transformation of x to the time domain is made by thelinear relation x=kt+x wherein i=0 is the instant of time defining thestart of the time interval referred to above. The size of the timeinterval is given by It must be noted that in each specific instancewhere the linear transformation to the time domain is accomplished thevariable t is limited in its range from zero to and 2x The distancebetween the projections on the 1 axis of the endpoints is dig-$ k sincethe abscissa of the lower limit of the function is t=0 and the abscissaof the upper limit of the function is The graph thus terminates verycertainly at points determined by the region of interest, although r,the independent variable, representing real time, of course continuesindefinitely and therefore =2(kt+x could ostensibly be plotted as a lineindefinitely long.

For purposes of this invention a regulatory repeated time representationof the function is required. This is called a periodic timerepresentation of the function. In general it is not practical to writean equation for a pcriodic time representation, although in specificcases it may be simple to do so. The equation above,

represents the actual equation of only one portion of one cycle of theperiodic representation, namely, that portion of one cycle whichexhibits the functional relationship between a dependent variable and anindependent variable exemplified by the equation =2x wherein at isconsidered to lie only between x and x and wherein, correspondingly,varies only from 2x to 2x FIG 1 shows one example of a periodic timerepresentation of the function =2x. This graph would be said torepresent the functional relation 5=2(kt+x in the region from i=0 to butit must be observed that in fact this functional relation holds only forthe segments from a to b, from d to e, from g to it, etc., and then onlyif t be regarded as starting at zero at each low tel-minus, i.e., at a,again at d, again at g, etc. The segments of the graph from b to c, fromc to d, from e to 1, from y to g, etc., are not represented by theequation =2(kt+x From the foregoing it is clear that a periodicrepresentation of a function involves displaying the functionrepetitively in time in such a manner that equal intervals of timecorresponds to the range of the indepedentent variable. Thus in FIG. '1,which illustrates a periodic time representation of the functionalrelation =2x 5. wherein x, the independent variable, ranges from x to xthe interval of time represented by the lengths ac, df, etc. correspondsto the range of x from x to x From the graph in FIG. 1 it is seen thatthe basic period of the graph, T, is represented by the lengths ad, dg,etc. As shown, only a portion of each basic period of the timerepresentation is occupied by the function, e.g., the time intervalsrepresented by abscissa lengths ac, d etc. The functional relation isnot being represented during a portion of each period shown as the timeinterval-s bcd, efg, e'tc., each of which has a duration The presentinvention can use periodic time representations of the type shown inFIG. 1 wherein the repeated representations of the functional relationof interest are separated by a line on the graph representing -a valueor values not essentially of interest. The invention can also useanother type of periodic time representation wherein the functionalrelation of interest effectively occupies the entire period of timeunder consideration. This other type of periodic time representationfalls in two categories: (1) Where the repeated representations of thefunctional relation of interest are contiguous, and (2) where thefunctional relation of interest is contiguous to and alternates with itsmirror image. This latter type of periodic time representation is themost common and the simplest to use and to understand in its behavior inthe practice of the invention. The former type, exemplified in FIG. 1,is sometimes more convenient to produce. An explanation of thegeneration and use of this former type in the invention is set forthhereinafter in relation to the embodiments of FIGS. 26 and 27.

The common term for a device which gives a periodic time representationof a function is a wave form generator. In contrast to this, the termfunction generator implies a device such that if a value of anindependent variable is introduced, the device produces thecorresponding functional value. The independent variable may or may notbe varying with time. If the Wave form generator produces a periodictime representation wherein each functional display follows itspredecessor immediately with no dead interval between them, the periodictime representation is said to have a 100% duty cycle. In FIG. 1 if theabscissa intervals cd, fg, etc., were each reduced to zero therepresentation would have a 100% duty cycle.'

The actual duty cycle of FIG. 1 is given by lcT A RELATION BETWEEN AFUNCTION AND ITS INVERSE Given a function f(x) defined for a x b let ydenote a particular but arbitrary value of the dependent variable y.Define the variable E as percent b I Brian:

and for n odd where A and A are quantities independent of the variablex. Obtain the average value of E over the range of x. With theappropriate choice of values for the constants A and A depending uponthe nature of f(x), it develops that for many functions of practicalimportance, the re lation E =kf- (y holds, with k indepedent of y As anillustration, let f(x) =x a xb, and define E as if y 3 a if 2/0 Then,

E 1 Ed o-(oh The following is concerned with establishing generalformulas for E for a large class of functions of practical importance.Interest thus centers upon the integral Then b Xom fEdx=f Edx=Evaluation of the first and last integrals presents no problem since theintegrands are either A or A However, the second integral requirescareful consideration of the nature of the set f (y Only the case ofprincipal interest, namely when the set f (y is finite, will beconsidered here. Accordingly, if f' (y has wcardinality n, let x denotethe ith value of x in the set f' (y with x =x and x =x Then f ndwf ndwfndw XUm 01 02 Xom XOM Un Edit Imposing the further restriction that forno x is f(x a maximum or minimum, then, with n even, the first and lastintegrals of (2) will have the same integrand values, that is, both A orboth A If n is odd, one will be A and the other A We may, therefore,Write for n even Collecting terms and simplifying, the general formulafor b I Eda;

can be written as and O e (x a). In the event that y is a maximum orminimum for one or more x then each such x must be treated as two pointswith a corresponding increase in the value of n. Formulas 4a through 4dwill then apply.

It may be noted that results in the several foregoing analyses would besubstantially the same if B were alternatively defined as 2 if y 2 go 1if y yo E= A if y y K if y'=y where K is a constant of any finite valueExamples Here f (y )=y and since there is only one real valued root fory n: 1. Further, f(x =f(x e) O for every x because x is an increasingfunction. Formula 40 is therefore applicable and leads to the expressionb L (A1 A2)$ +A2b A a. Dividing through by (b-a) to obtain E Letting A=b and A =a, then E ='x x /y Also, (x )f(x e) 0; hence, Formula 4b isapplicable, resulting in 1 ave 2 1) 01 02) 'i 2' Now using the fact thatx x and croosing A =a, 14 :0, leads to ave= o1 y0 y= sin x, 0 21r 8Since f- (y =arc sin y then 11:3 if y =0, and 11:2 if y O. It isrecalled here that values of x for which y is a maximum or minimum (wheny =il) are each to be treated as two points.

(a) For y 0, Relation 4a holds since x =1rx If we choose A =1r/2 and A=1r/2, then 01 (b) For y 0, Relation 4b is applicable. Hence since x=31rx Again letting A =1r/2 and A ='rr/2 as in case (a) =1r:E thesupplement of X01 (c) For :0, Relation 4c is used. Therefore Once again,letting A =1r/ 2, A =7r/ 2 and noting that x =O, x02 7f, x03=271, 'thfiresult is =0 as required The significant result in this example is thatby letting A =1r/2 and A =1r/2 for all three cases corresponding to y Oand y 0, the value of E in each case corresponds to a correct, butnumerically smallest member of the set are sin y One can thereforewrite:

This result is extensively employed in the section on illustrativeapplications.

Monotonic functions with single valued inverses are readily handled byFormula 4c if the function is increasing (Example 1) and by (4d) if itis decreasing. If the function f(x), a x b, is an increasing tone, itfollows from (40) that choice of A =b and A =a makes M onotonicfunctions b X034 1) I J; E d:z:= A dx+jg for a non-decreasing functionand to b XAM b J; Edx: a A dx+f for a nomincreasing function Hence f 1(M z( rnvr), f(a:) non-decreasing a 2( OM' 1( oM), fi non-increasing Iffor f(x) non-decreasing the choice A =b, A =a is made, and for f(x)non-increasing A a, A =b, then in either case E x =maximum member of theSet f o)- In the foregoing examples it was shown that the appropriatechoice of values for A and A in the variable E(y,y leads to an averageof this variable which is equal to the least of the inverse values f (yThe invention uses this mathematical principle for the following methodof generation of a function of a single variable f(x). By generation ofa function of a variable is meant the production of a physical quantitysuch as voltage, current, electrical resistance, mechanical displacementor the like whose magnitude varies in accordance with the variation ofthe function of the variable.

Method of junction generation Object: To generate y flx).

Step 1.-Generate a periodic time representation of the inverse functionf* (y).

Step 2.-Compare the amplitude of this time function with a given value xof the independent variable of the required function f(x).

Step 3.-Generate, as a result of Step 2, a discontinuous function wherethe values of A and A are time independent, or at least do not varyappreciably over a single period of 1 0)- Step 4 .-Take the time averageof E(x,x This time average, for appropriately chosen values of A and Ais proportional to the value of the dependent variable y correspondingto x=x it should be noted that the value of x, namely x is permitted tochange only at a rate which is much smaller than 1/ T, the repetitionrate of the periodic time represen-tation of the inverse function f-(y). Also, it should be noted that if the function f (y) happens to beinherently periodic, its period need not correspond identically with theperiod, T, chosen for the periodic time representation of f =(y) in themethod of this invention. For example if f (y)=sin 6, its period wouldnormally be regarded as 21r radians, constituting the length of theshortest equal sub-interval into which the range of the independentvariable, 0, can be divided and obtian exactly the same graph of thefunction in each sub-interval. However, in practicing the method of theinvention, wherein it is required to present a periodic timerepresentation of f- (y) =sin 0, which involves the substitution of(kt-H9 for t9, it is possible within the scope of the invention tochoose a period T for the function sin (kt-Hi corresponding to a rangeof 0 over only 1r radians. In such a case the periodic timerepresentation of f (y) would preferably be made up of a repetitivepresentation in regular sequence of only that generally S-shaped portionof the ordinary sine graph lying between 7r/2 and +1r/ 2.

GENERALIZATION OF BASIC METHOD OF FUNCTION GENERATION The symbolicexpression in Step 3 of the aforementioned method implies at firs-tblush that it is required to generate (1) E=A during the time interval,say A, throughout which x x and (2) E=A during the time interval, say 6,throughout which x x However, since Step 4 requires taking a timeaverage 16 of E, it should be apparent to those skilled in the art thatexactly the same end result will be obtained if (1) E is caused to havethe value A not during the time interval, A, wherein x xo, but during adifferent time interval, say A, so long as A=A; that is, so long as thelength of time in the interval A equals that in the interval A; and

(2) E is caused to have the value A not during the interval, 6, whereinx x but during a different time interval, say 6, so long as 5'=6.

Referring the explanation for simplicity to the occasion of a singletime representation of the inverse functional relation, the significantfact is that the tWo values A and A of E divide between them an intervalof time equal to the total length of time during which the timerepresentation of the inverse functional relation occurs. Actually, thegeneration of E need not even be simultaneous with the timerepresentation of the inverse functional relation although in practiceit is. The share of time interval assigned to A is equal to the lengthof time that x x and the remainder of the time interval is assigned to AHowever, it is totally immaterial to the value of the end. result,namely the time average of E, whether A takes its share from the firstportion of the time interval or from the last portion of the timeinterval or from the middle portion of the time interval or partly fromtwo or more such portions.

From the foregoing it is clear that the following is a-- Generalizedstatement of the method of function generation Object: To generatey=f(x).

Step 1.Generate a periodic time representation of the inverse functionf- (y).

Step 2.Compare the amplitude of this time function with a given value xof the independent variable of the required function f(x).

Step 3.-Generate, as a result of Step 2, a discontinuous function Aduring an interval of time equal to that when x$x A during an intervalof time equal to that when x 0 where the values of A and A are timeindependent, or at least do not vary appreciably over a single period ofStep 4.Take the time average of E(x,x This time average, forappropriately chosen values of A and A is proportional to the value ofthe dependent variable 31 corresponding to x=x It should be noted thatthe more extensively Verbalized expression E in Step 3 immediately aboveis fully equivalent to and interchangeable with the more succinct,predominantly symbolic expression in Step 3 of the earlier recitation ofthe method. Although the predominantly symbolic expression, being moreconvenient to write, will be generally used hereinafter, it must beunderstood and interpreted always to include the generalized expression.

FIG. 2 shows diagrammatically one apparatus of the present invention forcarrying out the aforedescribed method of function generation.

Numeral 2 indicates a generator of the periodic time representation ofx=f- (y). The output of this generator, being, for example, a voltage orthe like, represented by the expression x(t|-T), is fed into anamplitude comparator 4 into which is also fed the physical quantity suchas voltage, representing x the given value of x for which it is desiredto produce the corresponding value of the function of x. The amplitudecomparator 4 compares the value of x with the value of x generated bythe generator 2 as that value of x varies within the region of interestduring the time cycle. During the period of time when x, .the output ofgenerator 2, is less than or equal to x the amplitude comparator 4 putsout a first signal and during the time while the value of x fed into theamplitude comparator exceeds the value x the amplitude comparator putsout a second signal. The auxiliary function generator 6 generates thediscontinuous function E, which function has two values, one value beingproduced by the generator 6 when the generator 6 is receiving theaforementioned first signal from the amplitude comparator 4, and theother, when the generator 6 is receiving from .the amplitude comparator4 the aforementioned second signal. The output of the generator 6, whichagain may be an electrical quantity such as a voltage, is averaged by anaveraging device indicted by the numeral 8. When the voltages orcurrents are involved, such an averaging device can be constituted by afilter. The output of the averaging device 8 is simply the average valueof the auxiliary function E and represents, when the proper magnitudeshave been chosen for the two discrete values of E, the value y of thefunction of x corresponding to the value x of the independent variable.

Illustrative applications (1) Generation of x=% :1: To illustrate theuse of the method of this invention, let it be desired to generate thesimple function x= /z where 4 1 S Q S Q 2 and correspondingly 1$ X SMThe inverse function is =2x. Applying the method of the invention, aperiodic time representation of =2x is generated. One such periodic timerepresentation is shown in FIG. 3 which happens to have effectively a100% duty cycle. The amplitude of this time function is compared with agiven value of the independent variable of the required function x= /2.Thereupon there is generated, as a result of the comparison, adiscontinuous function In this example A is assigned the value x and Ais assigned the value x The auxiliary variable E over one cycle has thevalue x during the time interval OP and has the value x; during the timeinterval PQ. The time average of E is then taken over the cycle and thistime average will be the value x of the dependent variable x in theoriginal functional relation corresponding to In FIG. 3 the scale chosenat random happens to have the following values: x /2='1; x /2=4; 0P=2;PQ=6. Thus,-A :'4; A =1; and the time average over one cycle is givenby:

That is, x 1%. To check the validity of the method, a measurement of ismade and it is shown to be 3 /2, which fulfills the equation Thegeneralized concept of the basic method of the invention applied to thegeneration of x= /2 can be seen from the following. In FIG. 3, let therebe established on the t axis a point P located between P and Q, suchthat OP=PQ. Then, let the generation of the auxiliary variable takeplace in such a manner that E assumes the value A =x during the intervalPQ and assumes the value A =x during the interval OP. Since, under thisconcept, the two values A and A of the auxiliary variable E have dividedbetween them the total time interval OQ of the cycle of the timerepresentation of the inverse functional relation in the same proportionthat they did in the former case, when A =x occupied the interval OP andA =x occupied the interval PQ,

' then it is apparent that the average of E over the full cycle will beexactly the same as in the former case, and will equal x In thisinstance, E has the value A not during the interval of time, OP, whenbut during the interval of time PQ=OP. Similarly, E has the value A notduring the interval of time PQ when but during the interval OP=PQ. Inactual practice with electronic equipment, it is often more convenientto use an arrangement exemplified by this latter case, wherein A isgenerated during the interval PQ. This is particularly true when thetime representation of the inverse functional relation is symmetricalabout its intercept on the abscissa axis such as the sine time functionshown in FIG. 4. In such a case, the sum of the time representation ofthe inverse plus the given value of the independent variable changessign at the point corresponding to P and this change of sign is usefulto control the auxiliary function generator.

It is apparent that, in principle, the method of this invention can bepracticed by generating only a single cycle of the time representation=2(kt+x This will produce a precisely correct value x of the function x=/2 so long as remains fixed during the single cycle.

If remains fixed over a plurality of cycles of the time representation,the average of E over all these cycles will still be precisely x If E isaveraged over many cycles, say some thousands of cycles, it will remainindetectibly different from x even though the comparison of 5 with the 5of the time representation be caused to cease at some instant prior tothe exact completion of the last full cycle of the time representation.Since, in practice, it is commonly required to generate values of adependent variable corresponding to numerous values of an independentvariable it is, in practice, desirable to produce a periodic timerepresentation of, e.g., =2(kt+x so that there will always be at hand acontemporary cycle of this time function against which to compare anexisting value of 15 so as to generate promptly the auxiliary variable Eand hence the ultimately desired value x That is, the most usual case isthe one where p takes on various values as time progresses and does notremain fixed at one value.

If changes discontinuously to a new discrete value, say thecorresponding value x could be generated by merely generating oneadditional cycle of the time representation =2(kt+x and performing thecomparison and generation of E as in the first case. However, as justpreviously indicated, it would usually be desirable in conventionalcomputers to produce a periodic time representation, i.e., a continuousrepetition of the cycle, inasmuch as (15 usually will change with timeand, moreover, will usually change continuously with time. So long asthe value of 5,; remains substantially fixed during one cycle of thetime representation =2(kt+x the generated function will be substantiallyx Stated in other words, must for accuracy change at a much slower ratethan the repetition rate of the periodic time representation. If, forexample, were itself subject to a periodic variation, then, foraccuracy, the frequency of the variation of (p should be much less thanthat of the periodic time representation =2(lct+x In practice, if 1/ Tis the repetition rate of the periodic time representation, 1/100 thisrate or 1/ T is usually the maximum rate at which 5 will be allowed tochange to achieve practical computing accuracy. The slower the change inem, the more accurate will be the corresponding value of x that isproduced.

(2) Generation of 6=arc sin x The inverse function is x: sin 0. The sineis an inherently cyclic function with limiting values of +1 and 1. Aconvenient range for consideration of the function 6: are sin x is for1r/26 1r/2 since this corresponds to the range 1 l yielding a sampleextending over the entire possible range of the sine. The elementaryobvious segment of a sine curve to be used for exhibiting a periodictime representation of the inverse function x: sin 9 would be the regionwhere 6 ranges from 1r/2 13 to +1r/2 and the equation of one cycle ofsuch a representation would be x: sin (kt+ where 0 =1r/2 and 0 =1r/ 2 sothat t varies from The period of such a cycle is 1r/ k0=1r/ k. FIG. 4shows a periodic time representation of the sine function using theelementary segment from 1r/2 to +1r/ 2 as the basic constituent. Thegeneration of 0: are sin x for any given value x of x is accomplished inaccordance with the teaching of the invention viz. by comparing thissegmentary time representation over a cycle with x and generating theauxiliary function and then averaging E over the cycle. As mentioned inthe preceding illustrative application, the comparison and averaging canjust as well take place over a plurality of cycles of the timerepresentation and will give the same accurate result. Also, if xchanges with time, the only practical application of the invention is bythe use of a repetition of the cycle of the time representation and thisrepetition must for accuracy be at a rate much faster than the rate ofchange of x The generation of the wave form illustrated in FIG. 4,constituting a repetition of the segment of a conventional sine wavelying between 1r/2 and +1r/2, is certainly possible and can beaccomplished by methods well known in the art as explained, for example,in the volume entitled Waveforms, No. 19 of the Massachusetts Instituteof Technology Radiation Laboratory Series published in 1949 byMcGraw-Hill Book Co., New York. However, it is readily apparent thateach full cycle of such a wave form constitutes one symmetrical half ofthe conventional full sine wave cycle lying between 1r/2 and 31r/2. Itis further apparent that, because of the symmetry, the average value ofE obtained by comparison of x with that half of the conventional sinewave lying between 1r/2 and 31r/2 would be identical with that obtainedby comparison of x with the segment of a sine wave lying between 1r/2and 1r/2. Therefore it is clear the same identical accurate resultobtained by the use of the wave form of FIG. 4 can be achieved by usinga full sine wave form. The full sine wave form is easily generated bymeans of a sine wave oscillator and would normally be less expensive touse than the wave form of FIG. 4.

The use of the entire full wave output of an ordinary sine waveoscillator to generate 0': are sin x is now described. As previouslynoted, the inverse function is x: sin 0. Using conventional symbols, aperiodic time representation of the inverse function employing the fullwave is obtained by setting 0=wt, where t=time and w=angular frequency.The function x: sin wt is, as noted, easily generated by means of a sinewave oscillator. Next, the output of the sine wave oscillator iscompared with a given value of x, denoted by x and as a result of thiscomparison, there is generated the auxiliary function qr/ZifSiIl wtsx E(sm -i1r/2 if SID. wt a3 Because of the previously mentioned symmetry ofa sine wave, the average of E over one cycle of sin wt will be the sameas the average of E over that portion of the cycle lying between 0=1r/2and 0-:1r/2 and furthermore the average of E over one cycle will be thesame whether the cycle starts at x=-l or x=0 or elsewhere. Moreover, ifthe average of E is taken while x remains substantially unchanged duringmany cycles of sin wt, the value of E will be substantially the sameeven though the comparison of x with x is terminated before the exactcompletion of an integral number of cycles of sin wt.

Since E (sin wt, x is a periodic function of period Z1r/w, its timeaverage over a plurality of periods is the same as that over a singleperiod. This average has already, in effect, been obtained in Example 3above; and as before, there are three cases to consider: x O, x 0 and x=0. For x 0' we have (see FIG. 5):

1 t arc sin c =.wt =are sin as required The cases x 0 and x =0 aretreated in a similar man ner, all leading to the result that E =arc sinx 1r/2 E, 1r/2. Thus, by appropriate filtering of E(sin wt, x to obtainits time average, the value of arc sin x is generated. Since x was anarbitrary value of x within its range of definition, the function 0=arcsin x, 1r/2 01r/2 is obtained.

The physical schemes for carrying out the generation of 6=arc sin x, asin all the following examples, are very numerous depending on the natureof the variables and the speed and accuracy requirements. One suchscheme where the variables are voltages, as in electronic analoguecomputers, is shown schematically in FIG. 5. A voltage, representing sinwt, supplied by any convenient sine wave generator, is applied to theterminal 10 of an amplitude comparator 12. The amplitude comparator 12can be of any convenient form known in the art. Amplitude comparison andvarious types of comparators are described in the aforementioned volumewaveforms, especially in chapter 3 and chapter 9. A voltage representingx is supplied to terminal 14 of the comparator. The output of thecomparator 12 has two values: one if the compara-tor has found that x xand the other if x x The output of comparator 12 is fed to the generator16 of the auxiliary function E. The output of comparator 12 causesauxiliary function generator E to select one or the other of its twoinput voltages representing 1r/ 2 and 1r/2. It selects the former if x xand the latter if x x The output E of generator 16 is then adiscontinuous function having the two values constituted by the voltagesrepresenting 1r/2 and -1r/2. To average E this output is fed through alow pass filter, with cutoff below the frequency w, which effectivelytakes a time average of E so that the output at terminal 20 of thefilter 18 is E, which, as previously demonstrated, equals arc sin x Acompact electrical arrangement of the embodiment of FIG. 5 can be madeby joining together in one unit the comparator 12 and the auxiliaryfunction generator 16 wherein a polarized or differential lelay is used,operated by the combination of the voltage at 10 and the voltage at 14to make contact alternatively with a source of 1r/ 2 voltage or a sourceof 1r/2 voltage. Mechanical comparators embodying the invention includeany of the various forms of differential distance or angle detectorssuch as differential gears. Electronic comparators and switchingcircuits would preferably be used when the invention is used in a highspeed computer.

Although for simplicity of explanation the input to terminal 10 ofcomparator 12 was shown as sin wt, nevertheless in practice,particularly in conventional electronic computers, it is customary touse voltages of say volts to represent the limiting values of the rangeof a variable. Thus, more generally, the input at terminal 10 would beshown as say x=A sin wt where A might be 100 and A sin wt would be theactual instantaneous voltage at 10. In such a case -A x A. Similarly,the inputs at terminals 22 and 24 would more generally l5 be designatedas Ice/2 and -k1r/2. The actual voltage from filter 18 would then be E=.c arc sin (x /A). However, multiplying factors are as readily removedas inserted by conventional procedures and the actual value of thefunction can thus alwsy be extracted.

(3) Generation of 6=arc cos x Since arc cos x=arc sin x1r/2, it sufficesto add --rr/2 to the arc sine function in order to obtain the arc cosinefunction. This can be done in the circuit of FIG. 5 by adding 1r/2 tothe output of generator 16 or to the output of filter 18. If the arccosine function is desired it can readily be produced in theconventional manner known in the art by feeding +arc cosine into anoperational amplifier, the output of which will then be are cosine. Therange is 1r:0 0.

Of course 6=arc cos x can also be generated by the use of the method ofthe invention directly without recourse to a modification of the arcsine generator. This could be done by an apparatus similar to that ofFIG. 5 wherein the inputs to comparator 12 would be cos :2 and x and theinputs to generator 16 would be 11' and 0 instead of 1r/2 and -7r/2. Itshould be noted that cos wt is, of course, identical in form to sin ofand therefore is obtained from an ordinary sine wave oscillator, whichcan, as well, be called a cosine wave oscillator. The function thengenerated by generator 16 would be 0 if :cfix {1r if x zv This is forthe range 0 0 1.

(4) Generation of sin 0 and cos 0 This can be done for sin 0 in one oftwo ways: (A) by placing the arc sine circuit of PEG in the feedbackofan amplifier; or (B) by a direct application of the method of theinvention. Both methods are easily adapted to the generation of cos 0.Method A is illustrated in FIG. 6 and Method B is shown in FIG. 7.

In FIG. 6 numeral 25 designates an arc sine generator identical to theentire assembly of FIG. 5 which receives sin wl at one input terminal 26and receives y at its other input terminal 27 and yields arc sin y atits output terminal 28. The output of the arc sine generator, and avoltage representing 6, applied at input terminal 29, are each fedthrough separate identical resistors R to the summing junction 30 of anoperational amplifier 32. The output of this amplifier at 34 will be aquantity such that its arc sine equals +0, This quantity is then sin 0.This arrangement is operative in the region from 1r/2 to 1r/ 2. Bythrowing the switch 35 from the zero position to the position where 1r/2is fed into summing junction 30 through another resistor R, of the samevalue as each of the aforementioned two resistors, the output of theapparatus becomes sin ((911'/ 2) which equals -cos 0. If cos 6 isdesired, it is a simple matter to feed the output at 34 into anamplifier to reverse its sign. It should be noted that the range of thedevice of FIG. 6 when used to generate a cosine function is from 0 0. 1.

In FIG. 7 an apparatus using the direct application of the method ofthis invention is shown. A comparator 38 is supplied at terminal 4-0with z(t) a periodic time repre sentation of the arc sine function ofthe variety shown in FIG. 8, for example. The voltage representing 6,whose sine or cosine is ultimately to be produced, is fed into terminal42. The comparator compares the two voltages at terminals 4t) and 42 andthen zactuates auxiliary function generator 44, which is supplied withvoltages at terminals 46 and 48 representing +1 and 1, so that generator44 generates l1 if z(t) 6 The output of generator 44 is averaged byrunning it through a low pass filter 50 whose cutoff is below frequencyl/I bi l hig enough to have little effect on the maximum frequency ofchange of 6. The output of filter 59 at terminal 5'2 is then y sin 6where 1r/2 0 ,-./2. By throwing switch 54 from the zero terminal to the1r/2 terminal, the independent variable input to the comparator becomes01r/2 instead of 0 and the device will be made to produce y=sin(07r/2)=C0S 6 where O Q m As previously mentioned cos 0 can easily beconverted into cos 6 by feeding it through an amplifier.

The periodic time representation of the arc sine function can beobtained in a variety of ways for use in Method B. Among these are:

(a) Harmonic synthesis of time sine functions which is simply thereverse of harmonic or Fourier analysis;

(b) Harmonic modification of a square wave which amounts to filteringout from a square wave (which contains practically all frequencies) suchfrequencies that those which remain produce the desired time function;

(e) Letting the x input in FIG. 5 be .a triangular wave form ofamplitude +1 and +1 and of repetition rate much less than to. That is, xcan be varied as a triangular function of time and the output ofterminal 20 of such a device as FIG. 5 would then be a periodic timerepresent-ation of arc sin x (d) Direct modification of a periodic timefunction, such as sin wt, with a diode function generator.

The last mentioned item is shown in FIG. 8 Where sin wt is beingmodified to a time function that gives the values of the arc sinebetween 1r/2 and 1r/2 in a periodic manner. FIG. 9 shows the staticfunction that would have to be set up on a diode or similar functiongenerator to so modify sin ml. The use of diode function generators andthe like in this manner to accomplish modification of functions is fullyset forth in Korn and Korn op. cit. page 290 if.

As previously noted, in the illustrative sine and cosine generators ofFIGS. 6 and 7, the range of 6 is 1r/2 to 1r/2 for sin 9 and 0 to 1r forrcosine 0. These ranges can be extended by appropriate modification ofthe equipment when 0 exceeds these ranges. One example of an actualcircuit exhibiting such a modification is shown in FIG. 10. This circuitcan be said to represent essentially an actual circuit exemplifying theschematic arrangement of FIG. 6 plus the modification employed to extendthe range of B to from -31r/2 to 31r/2 for sin 6 and to 1r to 21: forcosine 6. The circuit comprises -a comparator including an operationalamplifier 54 having two input terminals 56 and 58 into which are fed,respectively, sin wt and y for comparison. The limiter connected toamplifier 54 is arranged to produce at the output terminal 60 adiscontinuous voltage function having only two values, say +2 if sinwt+y 0, and 2 if sin wt+y 0. This voltage is chosen as being sufficientto cause diode 62 either to conduct or not to conduct. The circuitfurther comprises an auxiliary function generator and an aver-agingdevice for its output including diodes 62 and 64, operational amplifier66 with input terminals 68 and 70, filter circuit 72 and output terminal74.

If sin wt y, the plate of diode 62 is made negative and therefore diode64 will conduct and the net voltage appearing at terminal 76 will bethat due to 1r/2 from terminal 68 minus, from terminal 70, ar/Zincreased by virtue of R /2 to 1r so that the net effect at terminal 76will be that of -1r/2. When sin wt y, the net voltage at terminal 76will be that due to effectively +7r/2. The output at 76 is averaged bythe filter circuit 72 so that arc sin y appears at terminal 74.

To produce the sine of 0, it suffices to embody the aforelescribed arcsine generator in the feedback of an amplifier circuit in the manner ofFIG. 6. In FIG. 10, the output 74 is placed in the feedback ofoperational amplifier 78, whose output at terminal 84) provides the y tobe fed into the arc sine generator at terminal 58. 9, Whose sine it isdesired to generate, has its negative applied at terminal 82 and joinsthe output of the arc sine generator at summing junction 84 serving asthe input source for amplifier 78. Since the entire monotonic section ofthe sine curve is represented by the portion lying between 6=1r/2 and=1r/2, the aforedescribed circuit will generate accurately the value ofsin for any 0 lying within these limits. As thus far described, theconstruction and operation of the circuit is substantially identicalwith that of FIG. 6. In the circuit of FIG. 6, and its counterpart inFIG. 10, if the value of the independent variable input 6 is allowed toexceed the limits 1r/2 and -1r/2, then the output of the device, i.e.,terminal 34 in FIG. 6 or terminal 80 of its counterpart in FIG. 10,would go very highly negative or positive until the amplifier saturatesand thus gives an erroneous reading. The reason for this erroneousreading is that the maximum voltage which the device, as thus fardescribed, can supply at terminal 74 in FIG. 10, for example, is -1r/2or +1r/2 and this is suflicient to balance at junction 84 only vr/Z or+1r/2 originating at terminal 82. 'If the difference between these twovoltages appearing at 84 is not very close to zero, the tremendousamplification of amplifier 78 causes its output at 80 to rise until theamplifier saturates.

To extend the limits of the function would require some modificationwhich would cause the output at terminal 80, which is, for example say+1 when 0 is 90, to decrease when 0 increases to, say 93, until itreaches the 'same value that it had when 0 was 87, since sin (90+3)=sin(903). In other Words, the circuit of FIG. 6 and its counterpart in FIG.10 can be made to produce a correct value for the sine of 0 with 0 equalto, say 93, if the effective 0 input were made 87 or in general if theeffective input were reduced to a value 62(01r/2). This is accomplishedin FIG. 10 by adding the two additional branches 86 and 88 to be usedunder appropriate circumstances to contribute to the voltage at terminal76.

The operation of the circuit can easily be understood by reference toFIG. 11 in which the solid line representation is a graph of effectiveinput to junction 84 in FIG. 10 versus 0, which latter is applied toterminal 82. As the input of 0 at terminal 82 goes from 1r/2 to 1r/2 theeffective input at junction 84 must go from 1r/2 to 1r/2 and it does so,as illustrated in FIG. 11 by the line segment PQ, by virtue of theoperation of the circuit heretofore described as the counterpart of FIG.6. As 0 increases beyond 1r/2 and the input 0 at terminal 82, designatedas 0 becomes more negative than 1r/2, it is required for the effectiveinput at 84, designated as 9 to decrease in absolute magnitude to thevalue given by the equation The reason for this can be seen from anexample using actual numbers. When say 9 =87, the output at terminal 80is sin 87. When 0 =90, the output at terminal 80 is sin 90. However, if0 should be allowed to become more negative to say 93, then the system,which is built to work only within the limits 1r/2 to 1r/2, cannothandle the 93 voltage and, so to speak, goes berserk yielding an outputat 80 representing saturation of amplifier 78. But, observing that sin93=sin 87, it is apparent that if, when 0 =93, 0 can be made equal to87", then the apparatus, which is fully capable of handling a voltage of87 at terminal 84 without going berserk, will yield at terminal 80 avoltage equal to sin 87. This latter, of course, is numerically equal tosin 93 so that the apparatus is, in effect, handling a voltage input at82 representing 0 1r/2.

It should be noted that the general requirement, previously stated, thatfor 0 1r/2, 0 must equal a2+ s2 is represented in the preceding numeralexample thus: 0 =93+2(93-90)=-87 terminal 84 upon To accomplish thisrequirement means contributing, at the time when 0 1r/ 2, a component at84 which will add, to the component at 84 due to s2, the effect of 2(01r/2) applied through an input resistor equal in size to 85. This addedcomponent arrives from the network comprised of branches 86 and 88. Thesame voltage -0 applied to terminal 82 is always simultaneously appliedto terminal 90. When 0 at terminal 90 is more negative than -1r/2, thepotential of the cathode of diode 94 is lowered below that of its plateand hence diode 94 conducts, causing a current to flow in branch 86whose magnitude is proportional to 0(1r/2) divided by R 2. This, ineffect, contributes at junction 76 a potential of 2(-0+1r/2) which, inpassing through amplifier 66, changes its sign and, since resistor 95equals resistor 85, appears at terminal 84 as, effectively, 2(6-1r/2),compared to the 0 at the same terminal contributed fr-omterminal 82. Thenet or effective input, then, at initiation of the operation is0+2(01r/2)=01r. If, as in the aforementioned example, 0:93, then the neteffective input at terminal 84 would correspond to 931r=87, a magnitudewhich is within the limits of 1r/2 to +1r/2 under which the circuit iscapable of giving correct results. The production of the propereffective input at terminal 84 for the region 1r/ 20 31r/2 is showngraphically in FIG. 11 by the dotted line segment PR, representing thecontribution from 0 the dash-dot line segment ST, representing' thecontribution from branch 86 equal to 2(0 1r/2); and the solid linesegment PU representing the sum of the two contributions at terminal 84.

An analogous situation occurs with conduction in branch 88 when 31r/2 01r/2. This is shown graphically in FIG. 11 by line segments QV and LMwhich add to produce QN.

This circuit can be used, by throwing switch 97 to the 1r/2 position,for generating cos 0 for the limits 1r 6 31r. But, of course, it isoperable only within these limits for the cos 6 (and -31r/2 931r/2 forthe sine) for the reason that these limits are necessary, with thiscircuit, to maintain the effective net input at 84 between 1r/2 and1r/Z. If 0 should exceed 31r/2, e.g., should be 271, then the neteffective input at 84 would be 01r=27l180=9l which is beyond theoperating limits of the circuit. However, further extension beyond therange 31r/2 to 31r/2 for the sine and n' to 21r for the cosine is, ofcourse, possible using the illustrated principle, i.e., by energizingappropriate circuits whenever the absolute magnitude of 0 exceeds 31r/2,51r/ 2, etc., so that the effective input at 84 is always maintained inthe range 1r/2 to +1r/ 2.

The circuit of FIG. 7 using Method B can also be modified to extend therange of 0. FIG. 12 illustrates such a modification showing oneparticular embodiment. When operating in the range of 1r/2 0 1r/2, thecircuit compares 0 applied at terminal 102 with the time representationz(t) of the arc sine function applied at terminal 104 and, on the basisof the comparison, selects, in a manner similar to the operation of thecircuit of FIG. 10, either +1 or 1 from terminal 106 or 108 as the valueof the auxiliary variable. The auxiliary variable is averaged by thefilter 110 yielding sin 0 at output terminal 112. If 0 exceeds Ir/2,diode 114 conducts and produces as the effective input at terminal 116the sum of 0-|2(0-1r/2)=01r, the first term on the left hand side beingdue to branch 118 and the second term being due to branch 120. This isso because resistor 119 is twice as large as resistor 121. So long as 031r/2, the quantity 01r effectively applied at 116 remains within the1r/2 to 1r/2 range of effective inputs within which the circuit givescorrect results. Similarly, when 0 1r/2 branch 122 conducts and thecircuit yields correct results for 6 31r/2. I'f switch 124 is swung tothe 1r/ 2 terminal the circuit operates to generate cos 0 for 1r 0 21r.As indicated in the discussion of FIG.

.19 10, the circuit of FIG. 12 can, of course, be extended using theillustrated principle beyond the range 31r/2 6 31r/2 for the sine and-1r r9 21r for the cosine.

Generation of sin 6, cos with unlimited angular range It is oftenimportant in problems using angles to have an unlimited angular rangewhen generating sine or cosine functions. The circuits of FIGS. 6, 7,10, and 12 can be adapted to this requirement through the use of anauxiliary circuit. This auxiliary circuit makes use of d0/dt to producean oscillation that sweeps through the restricted angular ranges of thesine and cosine generators (e.g., for one sine generator the range wouldbe from -1r/2 to +1r/2) at a rate proportional to dfi/dt. When dG/dt isconstant this oscillation becomes an isosceles triangular wave. Thecircuit, when used for example to supply a sine generator, will thensupply the sine generator with an input 6 which always lies between-1r/2 and 4-1/2 and at each instant has a value such that its sine isequal to the sine of the actual angle 6 (which is the actual machinevariable) at that instant. That is, the circuit in a sense performs afunction which results in the mathematical equivalent of converting theactual 0, no matter how large it may be, into an angle in either thefirst or fourth quadrants having an equivalent sine. The circuitperforms this function without receiving (after initiation of itsoperation) any actual 6 input but by receiving merely actual dO/dt inputwhich latter it integrates with respect to time in order to be able tosense increments of actual 6. A preferred embodiment of the auxiliarycircuit is shown in FIG. 13.

The circuit of FIG. 13 comprises an operational amplifier 126 Whoseoutput at terminal 128 will ultimately be the desired 0 whose negativewould be fed into, for example, terminal 29 of the sine generator ofFIG. 6 or the like. The amplifier 126 is shunted by a condenser 130. Thecapacitor-shunted amplifier 126, 130 is located in one branch 132 of aparallel circuit including another branch 134, which parallel circuit isconnected in series with a pair of operational amplifiers 136 and 138.Amplifier 136 is shunted by alternatively operating branch-es 140 and142, the former branch including a diode 144 and a voltage source suchas a battery 146 of value 7F/2, and the latter branch including a diode148 and a voltage source such as a battery 150 of such a value as toproduce at terminal 152 a voltage of 1r/2 when branch 142 is conducting.

Branch 132 includes two resistors 154 and 156 of equal value at Whosejunction 158 is connected the output of a circuit yielding angular rateof change. This angular rate circuit receives at its input terminal 160the quantity dfl/dt, the time rate of change of the actual machinevariable 9, which it can apply to junction 158 when diode 162 isconducting. Alternatively, when diode 164 is conducting, the angularrate circuit can apply d0/dt to junction 158, the negative beingobtained by simply passing dO/dt through the amplifier 166.

To initiate the operation of the circuit of FIG. 13, both 6 and dO/dtmust be initially available but, after initiation of the operation, allthat is needed is dB/dt and no further need exists for information as tothe value of the actual machine variable 0 to enable the device tocontinue functioning. The operation of the device proceeds as follows.At time t=0, 0, the quantity appearing at terminal 128, is assumed to be6(0). This value is established by applying, either automatically ormanually, a voltage across capacitor 130, this being the initial valueof the actual machine variable 6. This voltge can be applied by simplyplacing a battery of the correct value across the terminals of condenser130, it being remembered that the potential at the summing junction 168of the operational amplifier 126 is always substantially zero or ground.At the same instant that the initial value of 6 is applied acrosscondenser 130 a'H/dt is connected to terminal 160. At time t=0+6, thebattery imposing 6(0) across condenser 130 is removed. While the batterywas in position across condenser 130, the potential across thecondenser-was necessarily maintained constant. Upon removal of thebattery, however, the amplifier 126 with its associated condenser actsas an integrator and begins to integrate its input voltage which isapplied to one or both of its input resistors 154, 156. Assuming that 09(0) 1r/ 2, it will be intended for the integrator to add to the initialvalue 0(0) appearing at 128 the increment represented by the integral ofdfl/dt over a period of time until the value of 0 at 128 reaches 1r/2.To insure that the initial operation is started in the right directionto perform this addition, it is required that, at the start of theoperation, a positive input should exist at input terminal 188 to theamplifier 136. This can easily be accomplished by throwing the switch188 to a source such as 186 of positive potential, which could be forexample merely one volt, at the instant of the start of the operationand then throwing it back into the solid line position very rapidly,using a make-beforebreak switch if desired. The reason for applying aninitial positive potential at 188 can be seen from the followinganalysis.

With, say at 128 from the starting battery applied across 130, therewould be experienced at summing junction 174 the effect of +80 from 128plus the eflect transmitted from terminal 152. At 152 there will,however, be a voltage of -1r/2 produced by virtue of the followingsequence of events. When terminal 188 is connected to the positivebattery source 186, the output of amplifier 136 at 152 will be negative.By virtue .of battery 150 and diode 148, it is held at a negative levelof 1r/2. Therefore, at summing junction 174 there will be felt theeffect of, say, +80 from 128 combined with from 152 giving a netnegative effect at 174 which will emanate with a change of sign as apositive voltage at 172. This positive voltage at 172 is fed, throughresistor 189, into summing junction 170, thus maintaining the circuit ina stable state since the positive starting voltage at 188 from thebattery 186 was precisely the sign required to produce a positivevoltage at 172 to be fed into 188 so that the device will beself-maintaining.

With 1r/2 appearing at junction 152, as just described, the potential atjunction 158 will be -1r/4 since resistors 154 and 156 are equal and thepotential at 168, as previously indicated, is substantially zero. dG/dtis assumed to have a value between zero and 1r/4. The presence of -1r/4at junction 158 therefore causes diode 164 to conduct, thereuponclamping the voltage at junction 158 at the level of d6/dt which mightbe at say 40 volts. With a -40 volts at terminal 158, the integratingamplifier 126 will integrate this voltage continuously as long as it isapplied at terminal 158, resulting in an increase in the positivevoltage at terminal 178 and hence, at 128. When the voltage at 128 hasreached 1r/2-, a change will occur. As soon as the voltage at 128exceeds ever so slightly 1r/2, the net eifect at junction 174 will flipfrom negative to positive. For example, +91 at junction 128 combinedwith 1r/2 from junction 152 will yield a net effect at 174 of +1. Thispositive voltage at 174 changes its sign by passing through amplifier138, and the voltage at 172 will then be negative. A negative voltage at172 fed into junction 170 will produce a positive voltage at 152, whichpositive voltage will be fixed at 1r/ 2 by the limiting effect of branchhaving battery 146 and diode 144.

As soon as '+1r/2 appears at 152 this will tend to produce at junction158 a potential of -+1r/ 4 which instantly stops diode 164 fromconducting and causes diode 162 to conduct, transmitting to junction 158the voltage dB/dt originating at terminal 160. Assuming, as previouslystated, that d9/dt= +40, merely for example, this positive voltage at158 will appear as a negative voltage at

